The USQUE has been approved to be very attractive for the attitude estimation and has been extended to the integrated GPS and inertial navigation application. Actually, the USQUE can be used in any applications with quaternion such as the in-motion alignment. To calculate the propagated sigma points of the GRP in the USQUE the propagated sigma points of the error quaternion should be first determined, which is achieved by multiplying the propagated sigma points of the quaternion with a reference quaternion. In the USQUE the propagated sigma point of the quaternion in the center is selected as the reference quaternion. The intrinsic-gradient descent algorithm uses the fact that quaternion algebra provides a unique definition of the distance between two attitudes. The intrinsic-gradient-descent algorithm is an iterative method, and the number of iterations is usually very small.

A robust derivative-free algorithm named outliers robust unscented Kalman filter (ORUKF) is proposed to handle both the state and measurement outliers. Based on the generalized maximum likelihood perspective on the Kalman filter, the state is first augmented with the measurement noise, then the covariance of the augmented state is reformulated by the M estimate methodology and embedded into a modified version of the iterated unscented Kalman filter (UKF) to detect and suppress the outliers. Attractive features of the novel robust derivative-free algorithm include ability to handle multiple outliers, ability to exhibit the accuracy and flexibility of the UKF for the nonlinear problems, and high statistical efficiency under nominal conditions and flexibility to encompass maximum likelihood estimate.